Modular Networks and Nondeterministic Sequential Machines

The synthesis of sequential machines by interconnections of copies of a fixed module is considered. A family of modules Mr,pfor positive integers r and p, is defined. Mr,p can be used to synthesize sequential machines with 2^Pinput symbols. A nondeterministic sequential machine (NSM) is said to be r-bounded if it has one initial state and for no state and input are there more than r choices for the next state. It is shown that the problem of finding a network of modules Mr,prealizing a given event E is equivalent to finding an r-bounded NSM realizing the reverse of E. As a consequence, two upper bounds on the number of copies of the module Mr,pnecessary to realize an event E can be shown. 1) If E is defined by an n-state NSM, then E is defined by a network of at most cln^2+plogr²copies of Mr,p. 2) If E is defined by an n-state deterministic sequential machine, then E is defined by a network of at most c2n^l+vlogr²copies of Mr,p. cland C2are constants, about 4.